3.2. Uses of Heaps

There are two main uses of heaps.

The first is as a way of implementing a special kind of queue,
called a priority queue. Recall that in an ordinary queue, elements
are added at one end of the queue and removed from the other end, so
that the elements are removed in the same order they are added (FIFO).
In a priority queue, each element has a priority; when an element is
removed it must be the element on the queue with the highest priority.

A very efficient way to implement a priority queue is with a heap
ordered by priority - each node is higher priority than everything
below it. The highest priority element, then, is at the top of the
heap. We've just seen that the top element can be retrieved with a
single delete operation - O(logN) - and that inserting a new element
is also O(logN).

The second application is sorting. HeapSort uses the heap data
structure to sort values in exactly the same way as TreeSort used a
binary search tree. To sort an array, or list, containing N values
there are two steps:

insert each value into a heap (initially empty)

remove each value form the heap in ascending order (this is done
by N successive calls to get_smallest).

What is the complexity of the HeapSort algorithm?

(N insert operations) + (N delete operations)

Each insert and delete operation is O(logN) at the very worst -
the heap does not always have all N values in it. So, the complexity
is certainly no greater than O(NlogN). This is better than the
worst-case for TreeSort, which, because you might build a degenerate
binary search tree, is O(N*N).

HeapSort is especially useful for sorting arrays. Why? Because
heaps - unlike almost all other types of trees - are usually
implemented in arrays, not as linked data structures!